As a sidenote, "electrons don't rotate" is tricky.
As s orbitals, they don't.
As other orbitals, it's less simple. The probability density (hence the absolute value of the wavefunction) doesn't move, that's why such solutions are called "stationary", and the electron doesn't radiate. But the phase of the wavefunction does change over time and, at some angular moments, the phase rotates around the nucleus. This rotation defines an angular moment despite the solution is stationary.
And then, you have the wavefunctions for electrons that are trapped in atoms but are not orbitals. We know all wavefunctions for trapped electrons are linear combinations of orbitals, but these are not stationary: their absolute value moves over time. If you take [1s+2px]/sqrt(2) for instance, the electron has a bulge in its probability density, and if the 2p has an angular momentum of 0 the electron wobbles over the x axis, while an angular momentum of 1 lets the bulge rotate around the nucleus near to the yz plane.
This latter case resembles a classical rotation, just more fuzzy than a planet around a sun, and it does radiate or absorb an electromagnetic wave and is unstable.